U.S. Pat. No. 4,200,770, entitled "CRYPTOGRAPHIC APPARATUS AND METHOD," discloses a device for and method of performing a cryptographic key exchange over a public channel. The method is often called a public-key key exchange method or the Diffie-Hellman key exchange method after the first two named inventors of U.S. Pat. No. 4,200,770. U.S. Pat. No. 4,200,770 is hereby incorporated by reference into the specification of the present invention.
The cryptographic strength of the Diffie-Hellman key exchange method is based on the apparent intractability of finding a discrete logarithm, or discrete log, under certain conditions. Simply, two users exchange information by concealing their information using the mathematical technique of exponentiation. The users then mathematically combine the information they receive to the information they have to a key. The key may then be used with an encryption algorithm to encrypt a message. This method of establishing a private key between two users using a public channel solves the key distribution problem.
In order for an adversary to recover the concealed information and, therefore, be able to construct the key and decrypt messages sent between the two users, the adversary must be able to perform the inverse of exponentiation (i.e., a logarithm). There are mathematical methods for finding a discrete logarithm (e.g., the Number Field Sieve), but these algorithms cannot be done in any reasonable time using sophisticated computers if certain conditions are met during the construction of the key (e.g., the numbers involved in establishing the key are big enough).
More precisely, the cryptographic strength of the Diffie-Hellman key exchange method is based on the difficulty of computing discrete logs in a finite cyclic group. Mathematically, the discrete log problem is as follows. Let G be a finite cyclic group of order q, where g is a generator of G. Let r be a secret number such that 0&lt;r&lt;q. Given G, q, g, and g r, where " " denotes exponentiation, find r, where r is the discrete logarithm, or discrete log, of g r. The discrete log problem is to find r.
In a Diffie-Hellman key exchange, two users (e.g., User A and User B) agree on a common G, g, and q. User A generates, or acquires, a secret number "a", where 1&lt;a&lt;q, computes g a, and sends g a to User B. User B generates, or acquires, a secret number "b", where 1&lt;b&lt;q, computes g b, and sends g b to User A. User A then computes (g b) a, while User B computes (g a) b. Since these two values are mathematically equivalent, the two users are now in possession of the same secret number. A cryptographic key may then be derived from the secret number. The significance of this method is that a private key was established between two users by transmitting information over a public channel (i.e., an adversary sees the information being passed) but without knowing a or b, the key cannot be constructed from the information that is passed over the public channel. If the users keep "a" and "b" private and the numbers used to generate the key are large enough so that g (ab) cannot be mathematically derived from g a and g b then only the users know the key. In practice, the most common choice for G is the integers mod n, where n is an integer.
Large keys pose problems not only for the adversary but also for the users. Large keys require large amounts of computational power and require large amounts of time in order to generate and use the key. Cryptographers are always looking for ways to quickly generate the shortest keys possible that meet the cryptographic strength required to protect the encrypted message. The payoff for finding such a method is that cryptography can be done faster, cheaper, and in devices that do not have large amounts of computational power (e.g., hand-held smart-cards).
The choice of the group G is critical in a cryptographic system. The discrete log problem may be more difficult in one group and, therefore, cryptographically stronger than in another group, allowing the use of smaller parameters but maintaining the same level of security. Working with small numbers is easier than working with large numbers. Small numbers allow the cryptographic system to be higher performing (i.e., faster) and requires less storage. So, by choosing the right group, a user may be able to work with smaller numbers, make a faster cryptographic system, and get the same, or better, cryptographic strength than from another cryptographic system that uses larger numbers.
The classical choice for G in a Diffie-Hellman key exchange are integers mod n, where n is an integer as well. In 1985, Victor Miller and Neal Koblitz each suggested choosing G from elliptic curves. It is conjectured that choosing such a G allows the use of much smaller parameters, yet the discrete log problem using these groups is as difficult, or more difficult, than integer-based discrete log problems using larger numbers. This allows the users to generate a key that has the same, or better, cryptographic strength as a key generated from an integer G and is shorter than the integer-based key. Since shorter keys are easier to deal with, a cryptographic system based on a shorter key may be faster, cheaper, and implemented in computationally-restricted devices. So, an elliptic curve Diffie-Hellman key exchange method is an improvement over an integer-based Diffie-Hellman key exchange method.
More precisely, an elliptic curve is defined over a field F. An elliptic curve is the set of all ordered pairs (x,y) that satisfy a particular cubic equation over a field F, where x and y are each members of the field F. Each ordered pair is called a point on the elliptic curve. In addition to these points, there is another point 0 called the point at infinity. The infinity point is the additive identity (i.e., the infinity point plus any other point results in that other point). For cryptographic purposes, elliptic curves are typically chosen with F as the integers mod p for some large prime number p (i.e., F.sub.p) or as the field of 2 m elements (i.e., F.sub.2 m).
Multiplication or, more precisely, scalar multiplication is the dominant operation in elliptic curve cryptography. The speed at which multiplication can be done determines the performance of an elliptic curve method.
Multiplication of a point P on an elliptic curve by an integer k may be realized by a series of additions (i.e., kP=P+P+. . . +P, where the number of Ps is equal to k). This is very easy to implement in hardware since only an elliptic adder is required, but it is very inefficient. That is, the number of operations is equal to k which may be very large.
The classical approach to elliptic curve multiplication is a double and add approach. For example, if a user wishes to realize kP, where k=25 then 25 is first represented as a binary expansion of 25. That is, 25 is represented as a binary number 11001. Next, P is doubled a number of times equal to the number of bits in the binary expansion minus 1. For ease in generating an equation of the number of operations, the number of doubles is taken as m rather than m-1. The price for simplicity here is being off by 1. In this example, the doubles are 2P, 4P, 8P, and 16P. The doubles correspond to the bit locations in the binary expansion of 25 (i.e., 11001), except for the is bit. The doubles that correspond to bit locations that are 1s are then added along with P if the 1s bit is a 1. The number of adds equals the number of 1s in the binary expansion. In this example, there are three additions since there are three 1s in the binary expansion of 25 (i.e., 11001). So, 25P=16P+8P+P.
On average, there are m/2 1s in k. This results in m doubles and m/2 additions for a total of 3m/2 operations. Since the number of bits in k is always less than the value of k, the double and add approach requires fewer operations than does the addition method described above. Therefore, the double and add approach is more efficient (i.e., faster) than the addition approach.
While working on an elliptic curve allows smaller parameters relative to a modular arithmetic based system offering the same security, some of the efficiency advantage of smaller parameters is offset by the added complexity of doing arithmetic on an elliptic curve as opposed to ordinary modular arithmetic. For purposes of determining efficiency, elliptic doubles and elliptic additions are often grouped and considered elliptic operations. To gain even more efficiency advantages by going to elliptic curves, cryptographers seek ways to reduce the cost of an elliptic curve operation, or reduce the number of elliptic operations required. An elliptic curve method that requires fewer operations, or more efficiently executable operations, would result in an increase in the speed, or performance, of any device that implements such a method.
It is no more costly to do elliptic curve subtractions than it is to do elliptic curve additions. Therefore, a doubles and add approach to doing elliptic curve multiplication may be modified to include subtraction where appropriate. There are an infinite number of ways to represent an integer as a signed binary expansion. The negative is in a signed binary expansion indicate subtraction in a double/add/subtract method while the positive is in the signed binary expansion indicate addition in the double/add/subtract method. For example, 25 may be represented as an unsigned binary number 11001 (i.e., 16+8+1=25) or as one possible signed binary number "10-1001" (i.e., 32-8+1=25).
In an article entitled "Speeding Up The Computations On An Elliptic Curve Using Addition-Subtraction Chains", authored by Francois Morain and Jorge Olivos, published in Theoretical Informatics and Applications, Vol. 24, No. 6, 1990, pp. 531-544, the authors disclose an improvement to the double/add/subtract method mentioned above by placing a restriction on the signed binary expansion that results in fewer elliptic additions being required to do an elliptic curve multiplication and, therefore, increase the performance (i.e., speed) of elliptic curve multiplication. Messrs. Morain and Olivos proposed generating a signed binary expansion such that no two adjacent bit locations in the signed binary expansion are non-zero (i.e., two 1s, irrespective of polarity, may not be next to each other). Such a signed binary expansion is called a non-adjacent form (NAF) of a signed binary expansion. It has been shown that a NAF signed binary expansion is unique (i.e., each integer has only one NAF signed binary expansion) and contains the minimum number of 1s, irrespective of polarity. By minimizing the 1s, the number of additions is minimized. The improvement proposed by Messrs. Morain and Olivos still requires m doubles but only requires an average of m/3 additions for a total of 4m/3 elliptic curve operations. This is less than the 3m/2 elliptic curve operations required by the classical double and add method described above.
In an article entitled "CM-Curves With Good Cryptographic Properties", authored by Neal Koblitz, published in Crypto '91, 1991, pp. 279-287, the author discloses an improvement to the double/add/subtract method mentioned above by working in a particular family of elliptic curves (i.e., Koblitz Curves). Koblitz Curves are characteristic 2 curves of the form EQU E.sub.a :y 2+xy=x 3+ax 2+1, where a is a member of F.sub.2.
The group on which the key agreement is based is the group of F.sub.2 m-rational points on E.sub.a, which is chosen to have a low complexity normal basis. To operate on such curves, the multiplier k is expanded in powers of a complex number as follows: EQU .tau.=((-1) a+((-7) 0.5))/2.
The expansion is referred to as a base tau expansions. Similar to the binary expansions, the tau-adic expansion requires the analog of a double for each term in the expansion and an add for each non-zero term in the expansion. A property of these curves and normal basis representation is that the analog of doubling can be performed by a circular shift of bits and is, effectively, free. U.S. Pat. Nos. 4,567,600, entitled "METHOD AND APPARATUS FOR MAINTAINING THE PRIVACY OF DIGITAL MESSAGES CONVEYED BY PUBLIC TRANSMISSION," and 4,587,627, entitled "COMPUTATIONAL METHOD AND APPARATUS FOR FINITE FIELD ARITHMETIC," each disclose the method of getting the analog of doubles for free, but neither of these patents disclose the method of the present invention. U.S. Pat. Nos. 4,567,600 and 4,587,627 are each hereby incorporated by reference into the specification of the present invention. A downside of the base tau expansion is that it is 2 m-bits long for a k that is m-bits long. Another downside to the base tau expansion is that the rule for getting a minimum number of non-zero terms that was used in the binary case does not work for the base tau expansion. On average, 3/8 of the base tau expansion is non-zero. Since the base tau expansion is 2 m-bits long, the total number of elliptic curve operations is expected to be (3/8).times.2 m=3 m/4. This is less than the 4 m/3 elliptic curve operations required by the non-adjacent form (NAF) of the double/add/subtract method described above.
In an article entitled "Efficient Multiplication on Certain Nonsupersingular Elliptic Curves", authored by Willi Meier and Othmar Staffelbach, published in Crypto '92, 1992, pp. 333-343, the authors disclose an improvement to the base tau expansion described above. Messrs. Meier and Staffelbach disclose a method of generating a base tau expansion that is only m-bits long. They achieve this result by reducing k by mod(.tau..sup.m -1) and multiplying P by the (k mod(.tau..sup.m -1)). One-half of the terms of this reduced base tau expansion is non-zero. So, the expected number of elliptic curve operations for the reduced base tau expansion is mx(1/2)=m/2. This is less than the 3 m/4 elliptic curve operations required by the non-reduced base tau expansion method described above.
U.S. Pat. No. 5,159,632, entitled "METHOD AND APPARATUS FOR PUBLIC KEY EXCHANGE IN A CRYPTOGRAPHIC SYSTEM"; U.S. Pat. No. 5,271,061, entitled "METHOD AND APPARATUS FOR PUBLIC KEY EXCHANGE IN A CRYPTOGRAPHIC SYSTEM"; U.S. Pat. No. 5,272,755, entitled "PUBLIC KEY CRYPTOSYSTEM WITH AN ELLIPTIC CURVE"; U.S. Pat. No. 5,351,297, entitled "METHOD OF PRIVACY COMMUNICATION USING ELLIPTIC CURVES"; U.S. Pat. No. 5,463,690, entitled "METHOD AND APPARATUS FOR PUBLIC KEY EXCHANGE IN A CRYPTOGRAPHIC SYSTEM"; U.S. Pat. No. 5,737,424, entitled "METHOD AND SYSTEM FOR SECURE DISTRIBUTION OF PROTECTED DATA USING ELLIPTIC CURVE SYSTEMS"; and U.S. Pat. No. 5,761,305, entitled "KEY AGREEMENT AND TRANSPORT PROTOCOL WITH IMPLICIT SIGNATURES," each disclose a cryptographic method involving a key exchange method on an elliptic curve based on the discrete log problem, but none of these patents disclose a key exchange method that minimizes the number of elliptic curve operations as does the present invention. U.S. Pat. Nos. 5,159,632; 5,271,061; 5,272,755; 5,351,297; 5,463,690; 5,737,424; and 5,761,305 are hereby incorporated by reference into the specification of the present invention.
The present invention discloses an discrete log based key exchange method on an elliptic curve that requires fewer elliptic curve operations than the prior art methods listed above.